Optomechanical crystal with bound states in the continuum

Chipscale micro- and nano-optomechanical systems, hinging on the intangible radiation-pressure force, have shown their unique strength in sensing, signal transduction, and exploration of quantum physics with mechanical resonators. Optomechanical crystals, as one of the leading device platforms, enable simultaneous molding of the band structure of optical photons and microwave phonons with strong optomechanical coupling. Here, we demonstrate a new breed of optomechanical crystals in two-dimensional slab-on-substrate structures empowered by mechanical bound states in the continuum (BICs) at 8 GHz. We show symmetry-induced BIC emergence with optomechanical couplings up to g/2π ≈ 2.5 MHz per unit cell, on par with low-dimensional optomechanical crystals. Our work paves the way towards exploration of photon-phonon interaction beyond suspended microcavities, which might lead to new applications of optomechanics from phonon sensing to quantum transduction.


I. SYMMETRY ANALYSIS OF MECHANICAL BICS AND OPTOMECHANICAL COUPLING
The symmetry group of the stiffness tensor of silicon in the crystal plane and the hexagonal optomechanical crystal is C 4v and C 6v , respectively. For both C 4v and C 6v groups, there are two sets of equivalent mirror planes σ v and σ d as shown in Fig. S1. When one mirror plane of the silicon crystal lattice aligns with one mirror plane of the hexagonal optomechanical crystal, i.e., θ = 0 • , 15 • , 30 • , and 45 • , the mechanical system restores mirror symmetry and is described by C 2v group. Otherwise, the symmetry group for the mechanical system is C 2 .
The far-field acoustic plane wave propagating along the z−direction can be written as Q 0 = Q T,0 + Q L,0 = (ue x + ve y ) e ik 0 T ,z z + we z e ik 0 L,z z . (S1) According to Tables S1 and S2, e x and e y belong to B 1 and B 2 representations of C 2v and B representation of C 2 , respectively, and e z belongs to A 1 representation of C 2v and A representation of C 2 . Therefore, a mechanical mode at the Γ point can be a BIC, i.e., decouples from both transverse and longitudinal radiation waves, only if it belongs to the A 2 representation of C 2v group when θ = 0 • , 15 • , 30 • , or 45 • .
Next we analyze the optomechanical coupling of a unit cell based on the symmetry. The unit-cell optomechanical coupling is given by where m eff = u.c. ρ|Q| 2 dV is the unit-cell effective mass, ω m is the mechanical frequency. The moving-boundary and photoelastic components are calculated as and where ω o is the optical frequency,n is the normal vector of the interfaces, Q is displacement, S kl is strain tensor, the subscripts and ⊥ indicate the field components parallel and perpendicular to the surface and ∆ = int − ext and ∆ −1 = −1 int − −1 ext ( ext is the permittivity of the media whichn points to and int is the permittivity of the media on the other side).
For the moving boundary term, we write the numerator using a shorthanded notation u.c. (Q ·n)f dS. When there is mirror symmetry, this integration can be calculated as where χ Q (σ) stands for the character of mirror symmetry operation σ. Thus, the integral vanishes when the mechanical mode is odd with respect to the mirror plane. We have already used the fact that f (r) is even under the mirror operation as it is a quadratic function of E and D. This derivation is valid only when both mechanical and optical modes share the same mirror plane, which is true for θ = 0 • and θ = 45 • . For θ = 15 • and θ = 30 • , g OM,MB in general will be nonzero.
To calculate the photoelastic term, we note the photoelastic tensor for cubic crystal, such as silicon, in Voigt notation is given by   We find that if the mechanical mode is odd under either σ x or σ y , all the f ij s except f 16 are odd and their integrals are zero. For f 16 , because p 16 = 0 when θ = 0 • or 45 • , the term p 16 f 16 dV also vanishes. For θ = 15 • and 30 • , g OM,PE is in general nonzero.
In summary, A 2 modes for θ = 0 • and 45 • , which are BICs, have a zero optomechanical coupling. For other cases, optomechanical coupling is in general nonzero. This is also summarized in Table I of the main text.

II. OPTOMECHANICAL COUPLING OF FINITE OPTOMECHANICAL CRYSTALS
The optomechanical coupling for a sufficiently large optomechanical crystal can be calculated with the result of the unit-cell coupling. Performing the integrals of Eqs. S3 and S4 in the whole optomechanical crystal and apply the Bloch theorem, i.e., Q(r + R) = e ikm · R Q(r) and E(r + R) = e iko · R E(r), where k m and k o are the Bloch wavevector of the mechanical and optical modes, respectively, we havē where N c is the number of unit cells and R is the summation for all the lattice vectors. Same expression holds for g OM,PE . Because R e ikm · R is nonzero only when k m = 0, optomechanical coupling exists only for mechanical modes at the Γ point. Because the total massm eff = N c m eff , the optomechanical coupling of a sufficiently large optomechanical crystal is related to its unit-cell coupling as For finite optomechanical crystals, we have to consider the field envelop of the standing-wave resonances. The fields across unit cells now are related by Q(r + R) = e ikm · R φ (pm,qm) (R)Q(r) and E(r + R) = e iko · R φ (po,qo) (R)E(r), where φ (p,q) (R) is the envelop function of the (p, q)-th order standing-wave resonance. The envelope function can be approximately solved as the eigenfunction of a flat-top potential well within the boundary of the finite optomechanical crystal. Then we havē (S12) where the last integrals are performed in the whole optomechanical crystal. Similar expression applies to the photoelastic term. For the total effective mass, we havē Finally, the optomechanical coupling of a finite optomechanical crystal is related to its unit-cell coupling bȳ where the finite-size correction factor ζ is given by For the mechanical Bloch wavevector k m = 0, the numerically-calculated correction factor for a few standing-wave mechanical and optical resonances of a hexagonal crystal is listed in Table S5. The field envelops of some orders are shown in Fig. 3c. The order of the hexagonal crystal mode can be hard to identify. We determined it by adiabatically deforming the hexagonal crystal to a rectangular crystal and tracing the corresponding mode, where (p, q)-th resonance has p − 1 and q − 1 nodes along the x and y direction, respectively. The interaction Hamiltonian of the finite optomechanical crystal involving a pair of mechanical and optical standingwave resonances is given byĤ =ḡâ †â (b † +b), whereâ † (â) andb † (b) are the creation(annihilation) operators of the optical and mechanical resonances, respectively. When the optomechanical crystal is driven by a pump detuned from the optical resonance by the mechanical frequency, the interaction Hamiltonian could be linearized to beĤ = G(â †b † +âb) for blue-detuned pump orĤ = G(â †b +âb † ) for red-detuned pump, with the parametrically-enhanced optomechanical coupling, where n and n c = n/N c are the total photon number in the optical resonance and average photon number per unit cell, respectively.

III. MECHANICAL NOISE POWER SPECTRUM
The total noise power spectral density S measured by the photodetector is given by [1,2] where S e is the electronic noise of the detector, S EDFA is the noise of EDFA, S SN = √ P out ω l is the optical shot noise,, S m [ω] is the mechanical noise spectrum, κ = κ i + κ e is the total optical dissipation, G is the parametrically-enhanced optomechanical coupling, η is the total detection efficiency, G e is the detector gain factor from optical power to voltage, G EDFA is the EDFA gain, and R I is the input impedance of the spectrum analyzer. The optically-transduced mechanical noise spectrum is given by wheren m = k B T ωm the thermal occupation of the mechanical mode. The total detection efficiency is η = η cpl η t η det , where η cpl is the coupling efficiency of the grating coupler at the pump wavelength, η t is the total transmission efficiency in the optical fiber path, and η det is the quantum efficiency of the photodetector. We measured η t = 0.80 and η cpl ≈ 0.5 depending on the pump wavelength, while η det = 0.68 is given by the detector, which gives η ≈ 0.27.
We normalize the noise power spectrum with respect to the total noise floor which yields where we have used S EDFA ≡ G 2 e R I S EDFA and S SN ≡ G 2 e GEDFA R I S SN to denote the photodetector-measured EDFA noise and laser shot noise. The electronic noise S e is measured by blocking all the light. S SN + S e can be determined by removing the EDFA while keeping the laser power incident to the detector unchanged. Thus we obtain S SN and the ratio of . Finally, the measured normalized noise power spectrum is fitted using Eqs. S19 and S18, with ω m , γ, and G the only fitting parameters. The unit-cell optomechanical coupling is calculated from g = G/(ζ √ n c ), using n c = κ e P/((ω m ) 2 + (κ/2) 2 )/ 3 √ 3 2 N 2 and ζ depending on the optical standing-wave resonance that is deployed (see Table S5). For the four devices shown in Fig. 4a as an example, we used optical resonances of the following order, 0 • (1,1), 15 • (1,1), 30 • (1,1), 45 • (2,1), while the observed mechanical resonances are (1,1).

IV. FABRICATION RESULTS
The silicon-on-insulator microchip (220 nm silicon device layer and 3 µm buried oxide layer) is first rinsed with Acetone, isopropyl alcohol and deionized water to clean the surface. The pattern is then defined using electron beam lithography (Elionix ELS-G150) with 300 nm thick ZEP520A mask, followed by inductively coupled plasma reactive ion etch (Oxford PlasmaPro 100 Cobra) of silicon using SF6 and CHF3. The mask residual is removed by n-methyl-2-pyrrolidone hot bath and oxygen plasma cleaning (Diener Descum).
The SEM image of a typical unit cell is shown in Fig.S2a. The disorder of the snowflake patterns are measured both inside a same optomechanical crystal and across different optomechanical crystals (Table S6). The standard deviation of r and w is about 2.7% and 7.22% for the former and 2.6% and 6.7% for the latter. For comparison of different optomechanical crystals, we examined unit cells from the same location. Roughly speaking, the variation across different devices accounts for the the resonance frequency fluctuation and disorder within an optomechanical crystal induces scattering loss and decrease the resonance quality factor (see Section V).

V. SIMULATION OF SCATTERING LOSS
In multimode photonic/phononic crystals, structural disorders could cause scattering among different modes and thus introduce more loss channels to a given mode [3]. To estimate the quality factor of standing-wave mechanical resonances in disordered crystals, we simulated a 1 × 4 super-cell structure (Fig. S3a), where variations of both length (∆r) and width (∆w) of each snowflake hole are introduced. Each ∆r and ∆w are independently sampled and the standard deviation of ∆r and ∆w are controlled to be 4 nm and 2 nm, respectively, in accordance with the fabricated devices. We run 50 sets of simulations each for θ = 0 • , 15 • , 30 • , and 45 • . We set the Bloch wavevector to be k = 2π/(3aN ) with N = 25, approximately corresponding to the (1, 1) standing-wave resonance in size N = 25 hexagonal optomechanical crystals. The simulated radiative quality factors Q r combined with room-temperature material absorption Q a , which we assumed to be 3000, gives the total mechanical quality factor: 1/Q m = 1/Q r +1/Q a . The result is plotted in Fig. S3b. Due to the small size of the optomechanical crystal, there are no significant variations of Q m among four different θ's. We use the data set of all four angles here to calculate the upper and lower bounds of Q m as shown in Fig. 4.  Fig. S4a shows the fiber-optic coupler for the measurement of the optomechanical crystal. Light is guided by an angle-polished single-mode fiber, reflects at the polished surface, and is incident onto a grating coupler. The angle α of the fiber is chosen to ensure total internal reflection. Because of the inevitable air gap between the fiber and grating coupler, the incident angle onto the grating coupler is given by β = arcsin(n fiber · sin(90 • − 2α)). In general, the grating coupler is designed to convert the incident light to a guided wave by satisfying the phase-matching condition 2πn eff λ = 2π λ sin β + 2π Λ , where n eff is the effective refractive index of the grating and Λ is the lattice constant of the grating. It turns out an apodized grating coupler could yield higher coupling efficiency than a uniform grating coupler because of the better mode-matching with the single-mode optical fiber [4]. Thus, we adopt the apodized grating coupler scheme. The apodized grating coupler consists of 20 unit-cells with 40 individual tunable parameters l i and g i . The parameters are optimized using a standard inverse design approach [5] to maximize the coupling efficiency for different fiber angles. The optimized fiber angle is found to be 35.5

VI. DESIGN OF GRATING COUPLER
• and the corresponding l i and g i are shown in Fig. S4b. Fig. S4c shows the simulated transmission and measured device spectrum (inferred from the reflection spectrum). The peak efficiency is observed to be 52% with a 3-dB bandwidth of 40 nm.   S5a and b show the full mechanical and optical bandstructure of the nominal optomechanical crystal. The photonic crystal mirror consists of a triangular lattice of cylindrical holes with r = 78 nm and the same lattice constant a = 389 nm as the snowflake optomechanical crystal. The simulated band structure with the two lowest TE-like bands (Fig. S5c) shows a complete bandgap from 187 THz (1606 nm) to 201 THz (1493 nm). The photonic crystal mirror effectively suppresses the side-leakage but also perturbs the tail of the standing-wave resonance, inducing wavevector components that fall inside the continuum and causing out-of-plane radiation. We could alleviate such out-of-plane radiation by tuning the separation d between the photonic crystal mirror and optomechanical crystal [6,7]. To simulate this, we take a strip of optomechanical crystal of 25 unit cells as one super-cell and add periodic boundary conditions on the transverse direction, as shown in Fig. S5d. A sufficient number of photonic crystal mirror unit-cells are included and so the total quality factor Q t of the resonance is dominated by the out-of-plane radiation. Fig. S5e shows Q t ≡ Re [ωo] Im [ωo] of the fundamental standing-wave resonance for different d. An optimal d = 319 nm is observed, corresponding to a displacement of the photonic crystal from the nominal lattice constant by d − a = −70 nm.
The fabricated optomechanical crystal connects to a waveguide and the external coupling rate could be controlled by the number of phtonic crystal mirror layers between the waveguide and optomechanical crystal. In the simulation, we change the number of phtonic crystal mirror layers N m on one side (the waveguide side) of the optomechanical crystal super-cell while keeping the other side with a sufficient amount of photonic crystal mirror layers. We have also included four additional tapered mirror layers on the waveguide side. The external and internal quality factors are inferred from the energy flux as Q e = P e + P i,u + P i,d P e Q t , Q i = P e + P i,u + P i,d where P e is the energy flux towards the waveguide and P i,u and P i,d are out-of-plane energy flux upward and downward, respectively. Fig. S5f shows κ e /κ i ≡ Q i /Q e . The simulation is done for two cases. One case is with a large number of photonic crystal mirrors on the other side of the super-cell such that Q i ≈ 10 6 . In the other case, we purposely reduced the number of photonic crystal mirrors on the other side of the super-cell such that Q i ≈ 2 × 10 4 to mimic the fabricated optomechanical crystal. For this case. We find the critical coupling κ e /κ i ≈ 1 is achieved around N m = 4.